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Some Open Problems in the Theory of Subnormal Operators

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Some Open Problems in the Theory of Subnormal Operators Holomorphic Spaces MSRI Publications Volume 33, 1998 Some Open Problems in the Theory of Subnormal Operators JOHN B. CONWAY AND LIMING YANG Abstract. Subnormal operators arise naturally in complex function the- ory, differential geometry, potential theory, and approximation theory, and their study has rich applications in many areas of applied sciences as well as in pure mathematics. We discuss here some research problems concern- ing the structure of such operators: subnormal operators with finite-rank self-commutator, connections with quadrature domains, invariant subspace structure, and some approximation problems related to the theory. We also present some possible approaches for the solution of these problems. Introduction A bounded linear operator S on a separable Hilbert space H is called subnor- mal if there exists a normal operator N on a Hilbert space K containing H such that NH ⊂ H and NjH = S. The operator S is called cyclic if there exists an x in H such that H = closfp(S)x : p is a polynomialg; and is called rationally cyclic if there exists an x such that H = closfr(S)x : r is a rational function with poles off σ(S)g: The operator S is pure if S has no normal summand and is irreducible if S is not unitarily equivalent to a direct sum of two nonzero operators. The theory of subnormal operators provides rich applications in many areas, since many natural operators that arise in complex function theory, differential geometry, potential theory, and approximation theory are subnormal operators. Many deep results have been obtained since Halmos introduced the concept of a subnormal operator. In particular, Thomson's solution of the long-standing problem on the existence of bounded point evaluations reveals a structure theory of cyclic subnormal operators. Thomson's work answers many questions that had Yang was partially supported by NSF grant DMS-9401234. 201 202 JOHN B. CONWAY AND LIMING YANG been open for a long time and promises to enable researchers to answer many more; see [Thomson 1991] or [Conway 1991]. The latter is a general reference for the theory of subnormal operators. Here we will present some research problems on subnormal operators and discuss some possibilities for their solution. 1. Subnormal Operators with Finite-Rank Self-Commutator Let A denote area measure in the complex plane C. A bounded domain G is a quadrature domain if there exist points z1; : : : ; zN in G and constants am;n such that N Nn f(z) dA = a f (m)(z ) Z m;n n G nX=1 mX=0 for every function f analytic in G that is area-integrable. The theory of quad- rature domains has been successfully studied by the techniques of compact Rie- mann surfaces, complex analysis and potential theory [Aharonov and Shapiro 1976; Gustafsson 1983; Sakai 1988]. The self-commutator of S is the operator [S∗; S] = S∗S − SS∗. The structure of subnormal operators with finite-rank self-commutator has been studied by many authors. Morrel [1973/74] showed that every subnormal operator with rank one self-commutator is a linear combination of the unilateral shift and the identity. Olin et al. [≥ 1997] classified all cyclic subnormal operators with finite-rank self-commutator. D. Xia [1987a; 1987b] attempted to classify all subnormal operators with finite-rank self-commutator. His results, however, are incomplete. In [McCarthy and Yang 1997; 1995] a connection between a class of subnormal operators with finite-rank self-commutator and quadrature domains was established. However, the following problem still remains open. Problem 1.1. Classify all subnormal operators whose self-commutator has finite rank. One can show that, if S is a pure subnormal operator with a finite rank self- commutator, the spectrum of the minimal normal extension N is contained in an algebraic curve. This is a modification of a result in [Xia 1987a]. If one assumes some additional conditions|for example, that the index of S − λ is constant| then it turns out that S is unitarily equivalent to the direct sum of the bundle shifts over quadrature domains introduced in [Abrahamse and Douglas 1976]. (The argument is similar to that in [Putinar 1996], and uses the fact that the principle function is a constant on the spectrum minus the essential spectrum). For the general case, where the index of the subnormal operator S − λ may change from one component to another, difficulties remain. Example 1.2. Set r(z) = z(2z − 1)=(z + 2) and let Ω = r(D), where D is the open unit disk. If Γ1 = @Ω and Γ2 = clos r(@D) n @Ω , then Γ2 is a closed SOME OPEN PROBLEMS IN THE THEORY OF SUBNORMAL OPERATORS 203 simple curve. Let Ω0 denote the region bounded by Γ2 and let Tr be the Toeplitz 2 operator on H with symbol r. It is easy to show that Tr is a subnormal operator with a finite-rank self-commutator and that ind(Tr − λ) = −2 for λ 2 Ω0 and ind(Tr − λ) = −1 for λ 2 Ω n Ω¯ 0. Obviously, neither Ω nor Ω0 is a quadrature domain. The techniques in [McCarthy and Yang 1997] do not work for regions that have inner curves like Γ2. In [McCarthy and Yang 1997] one sees that the spectral pictures of rationally cyclic subnormal operators determine their structure. Therefore, in general, the difficulty consists of using the spectral picture of a subnormal operator with finite-rank self-commutator to obtain its structure. Example 1.3. Use the notation of Example 1.2. Let S be an irreducible sub- normal operator satisfying these properties: (1) σ(S) = Ω.¯ (2) σe(S) = Γ1 [ Γ2. (3) ind(S − λ) = −1 for λ 2 Ω n Ω¯ 0 and ind(S − λ) = −3 for λ 2 Ω0. The existence of such an irreducible subnormal operator is established by a method in [Thomson and Yang 1995]. Such an operator should not have a finite- rank self-commutator. However, how can one prove it? Understanding these examples will give some ideas on how to solve the problem. Recently M. Putinar [1996] found another connection between operator theory and the theory of quadrature domains, by studying hyponormal operators with rank one self-commutator. It is of interest to see how operator-theoretical meth- ods can be applied to the theory of quadrature domains. Let G be a quadrature domain and let ! be the harmonic measure of G with respect to a fixed point in G. Let H! be the operator of multiplication by z on 2 the Hardy space H (G). It is easy to see that H! is irreducible with a finite-rank self-commutator and that the spectrum of the minimal normal extension N! is @G. Therefore, one sees that @G is a subset of an irreducible algebraic curve (the spectrum of N! is a subset of an algebraic curve). This was proved by A. Aharanov and H. Shapiro [1976] using function theory in 1976. B. Gustafsson [1983] obtained even better results: he showed that, except for possibly finitely many points, the boundary of a quadrature domain is an irreducible algebraic curve. In order to prove Gustafsson's theorem by using operator theory, one needs to prove the following operator-theoretical conjecture. Conjecture 1.4. If S is an irreducible subnormal operator with finite-rank self- commutator, then except for possibly a finite number of points, the spectrum of the minimal normal extension N of S is equal to an irreducible algebraic curve. A solution to this should indicate some ideas for obtaining the structure of sub- normal operators with finite rank self-commutator. 204 JOHN B. CONWAY AND LIMING YANG Let G be a quadrature domain of connectivity t + 1 and order n. Let W be a plane domain bounded by finitely many smooth curves and conformally equivalent to G. Let φ : W ! G be a conformal map. Define 0 C = z 2 @G : z = φ(w) for some w 2 @W with φ (w) = 0 ; D = z 2 @G : z = φ(w1) = φ(w2) for two different wj 2 @W : It was shown in [Aharonov and Shapiro 1976; Gustafsson 1983] that, if G is a quadrature domain, there is a meromorphic function S(z) on G that is continuous on G¯ except at its poles and such that S(z) = z¯ on @G; moreover the equation S(z) = z¯ has at most finitely many solutions in G. Let E denote the set of the solutions in G. Let c, d, e denote the cardinalities of C, D, E. Theorem 1.5 [Gustafsson 1988]. t + c + 2d + e ≤ (n − 1)2. Theorem 1.6 [Sakai 1988]. c + e ≥ t + n − 1. Using the model for rationally cyclic subnormal operators with finite rank self- commutator, it has only been possible to show that t + e ≤ (n − 1)2 [McCarthy and Yang 1997]. Problem 1.7. Use operator-theoretical methods to prove Gustafsson's theorem and Sakai's theorem. In order to solve the problem, one has to understand deeply the operator- theoretical meanings of the numbers t, n, c, d, e. Discovering such methods may improve the results in the theory of quadrature domains and will help us understand the connections between these two areas better. 2. Invariant Subspaces of Subnormal Operators Let S be a cyclic subnormal operator. A standard result [Conway 1991, p. 52] says that S is unitarily equivalent to the operator Sµ of multiplication by z on the space P 2(µ), the closure of the polynomials in L2(µ). A point λ 2 C is a bounded point evaluation for P 2(µ) if there exists C > 0 such that jp(λ)j ≤ Ckpk for every polynomial p.
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